Transits and Occultations of an Earth-Sized Planet in an 8.5-Hour Orbit


We report the discovery of an Earth-sized planet () in an 8.5-hour orbit around a late G-type star (KIC 8435766, Kepler-78). The object was identified in a search for short-period planets in the Kepler database and confirmed to be a transiting planet (as opposed to an eclipsing stellar system) through the absence of ellipsoidal light variations or substantial radial-velocity variations. The unusually short orbital period and the relative brightness of the host star ( = 11.5) enable robust detections of the changing illumination of the visible hemisphere of the planet, as well as the occultations of the planet by the star. We interpret these signals as representing a combination of reflected and reprocessed light, with the highest planet dayside temperature in the range of 2300 K to 3100 K. Follow-up spectroscopy combined with finer sampling photometric observations will further pin down the system parameters and may even yield the mass of the planet.

Subject headings:
planetary systems—planets and satellites: detection, atmospheres

1. Introduction

The work described here was motivated by our curiosity about planets with the shortest possible orbital periods. Although many exoplanets have been discovered with orbital periods of a few days—most famously the “hot Jupiters”—relatively few are known with periods shorter than one day. Howard et al. (2012) found that such planets are less common than planets with periods of 2–3 days, based on data from the Kepler spacecraft. The shortest-period transit candidate is the 4.5-hour signal in the KOI 1843 system found by Ofir & Dreizler (2012), although this candidate has not yet been thoroughly vetted. Among the well-documented planets the record holder is Kepler-42c, with a period of 10.9 hr (Muirhead et al. 2012). The planet 55 Cnc e is the shortest-period planet (17.8 hr) for which the radius and mass have both been measured (Winn et al. 2011; Demory et al. 2011). In all of these cases the planet is smaller than about 2 . Among giant planets, the shortest period belongs to WASP-19b ( hr, Hebb et al. 2010).

The rarity of giant planets with  day could reflect the vulnerability of such planets to tidally-induced decay of their orbits (see, e.g., Schlaufman et al. 2010 for specific predictions), a possible tidal-inflation instability (Gu et al. 2003a), Roche-lobe overflow (Gu et al. 2003b), or evaporation (see, e.g., Murray-Clay et al. 2009). If so, then because smaller rocky planets are less vulnerable to these effects, one would expect smaller planets to be more common than large planets at the shortest periods. A suggestion that this is indeed the case comes from perusing the list of the active Kepler Objects of Interest (KOI)12. In this list we find only 17 active planet candidates with  hr, all of which have inferred planet sizes smaller than that of Neptune. However, this result is difficult to interpret because of the possibility of false positives due to eclipsing binary stars. There are already 55 known false positives in that period range, and the vetting is incomplete for most of the 17 candidates that remain.

There is also the possibility that the KOI catalog is missing some objects with short periods or short transit durations, as noted by Fressin et al. (2013). Other authors have performed independent searches of the Kepler database using the Box Least Squares (BLS) algorithm (Kovács, Zucker, & Mazeh 2002) and found new candidates (Ofir & Dreizler 2012; Huang, Bakos, & Hartman 2013). The BLS algorithm is designed for finding transit signals of short duration compared to the orbital period. For the shortest-period planets, though, the transit duration is a sizable fraction of the orbital period, and a Fourier Transform (FT) analysis should be sufficient for detection. The FT has the advantages of simplicity and speed. This was the technique used to detect the apparently disintegrating planet KIC 12557548b, with an orbital period of 15.6 hr (Rappaport et al. 2012).

Here we describe an Earth-sized planet with an orbital period of 8.5 hr, which was not among the Kepler Objects of Interest, and was identified in our FT-based survey of the Kepler data. Because so many transits have been observed and the star is unusually bright (with Kepler magnitude 11.5), the process of validation is simplified, the planetary occultations and illumination variations are easily detected, and further observations should be rewarding.

Section 2 of this paper describes the initial detection of the signal, our follow-up ground-based spectroscopic observations, and the properties of the parent star. Section 3 presents the analysis of the Kepler light curve and the determination of the system parameters. Section 4 demonstrates that the signal almost certainly arises from a transiting planet, as opposed to an eclipsing binary star, based on the lack of detectable ellipsoidal light variations or radial-velocity variations. Section 5 discusses the possibilities for the surface temperature and reflectivity of the planet, based on the observed properties of the illumination curve and occultations. We end with a brief discussion of the future prospects for studying the shortest-period planets.

2. Observations

2.1. Initial detection

To carry out an independent search for the shortest-period planets, we subjected all the Kepler long-cadence light curves to a FT analysis using data from Quarters 1-14. The light curves used for this study had all been processed with the PDC-MAP algorithm (Stumpe et al. 2012; Smith et al. 2012), which removes many of the instrumental artifacts from the time series while retaining the bulk of the astrophysical variability. The time series from each quarter was divided by its median value. Then, the normalized data from all quarters were stitched together into a single file. The FT was then calculated. We searched for the presence of at least one peak more than 4.6 standard deviations above the local noise level in the Fourier spectrum. To be considered further, we also required that the FT exhibit at least one harmonic or subharmonic that stands out at the 3.3 level. The candidates were then examined by eye. Only those that showed several harmonics with a slow falloff in amplitude with increasing frequency, and no sign of stellar pulsations, were selected for further study.

The surviving candidates underwent a period-folding analysis. To remove the slow flux variations caused by starspots and stellar rotation, we applied a moving-mean filter to the flux series, with a width in time equal to the candidate orbital period. Then we folded the time series with that period and inspected the resulting light curve, looking for the characteristic shape of a transit. We applied the same filtering algorithm to the time series of the row and column positions of the image photocenter (MOM_CENTR) provided by the Kepler pipeline. Systems with large photocenter shifts were discarded; such large shifts indicate that the flux variations belong to a neighboring star and not the intended Kepler target. We also checked for any significant differences in the depths of the odd- and even-numbered transits, which would reveal the candidate to be an eclipsing binary with twice the nominal period. A list of 20 potentially new short-period planet candidates passed all these tests, with orbital periods between 4 and 16 hours. As expected, this list is comprised entirely of objects smaller than Neptune. We will report on the entire collection in a separate paper. For this initial report we chose to focus on KIC 8435766 (from now on designated Kepler-78b), because it has the brightest host star, one of the shortest orbital periods, and the most significant detection of the illumination curve and occultations.

Figure 1 shows the time series, FT, and folded light curve for Kepler-78. The time series exhibits quasiperiodic flux variations with an amplitude of a few percent and a period of 12.5 days, likely the result of spots on a rotating star. The FT also shows a base frequency at  cycles day, and at least 9 higher harmonics, two of which are aliases resulting from reflection about the Nyquist limit of 25 cycles day. No subharmonics of these frequencies are seen; a positive detection would have been indicative of a binary with primary and secondary eclipses of nearly equal depth. The folded light curve shows a transit with a depth of 220 ppm and an occultation with a depth of 10 ppm. The illumination curve—the rise in flux between transit and occultation—is less obvious; see Figure 3 for a better view.

\epsfxsize=7in \epsffilefig1.eps

Figure 1.— Kepler data for Kepler-78 (KIC 8435766). Top.—Time series, based on quarterly-normalized PDC-MAP data. The quasiperiodic flux variations are characteristic of starspots and rotation. The right panel focuses on one arbitrarily chosen day of data, to allow several transits to be seen by eye (single 30 minute bins). Lower left panel.—Fourier transform, before any filtering of the starspot-induced signal. Lower right panel.—Light curve, after filtering and then folding the time series with a period of 8.5201795 hr.

2.2. Spectroscopy

Spectroscopic observations were undertaken to characterize the host star and to search for radial-velocity variations. We obtained five spectra with the fiber-fed Tillinghast Reflector Échelle Spectrograph on the 1.5m Tillinghast Reflector at the Fred Lawrence Whipple Observatory on Mt. Hopkins, Arizona. The observations took place in 2013 on March 23, 25, and 29, and on April 2 and 4. Individual exposure times of about 15 minutes yielded a signal-to-noise ratio (S/N) per resolution element in the Mg I b order ranging from 26 to 44, depending mostly on the seeing and sky transparency.

Spectroscopic parameters for the host star were determined with the Stellar Parameter Classification code (SPC; Buchhave et al. 2012). We derived the parameters from each spectrum, and then computed weighted averages. The results were  K, , , and . We also estimated the radial velocity of the star at .

To estimate the stellar mass and radius, we used the calibrated relationships provided by Torres & Andersen (2010) between the spectroscopic parameters and stellar dimensions. These relationships give a stellar mass , radius , and mean density g cm. Error propagation was performed assuming independent Gaussian errors in the stellar parameters, along with a 6% systematic error in the stellar mass due to the uncertainty in the calibration formulas. The final values are summarized in Table 2. Following Torres et al. (2012) we checked the derived mass and the radius by comparing the spectroscopic parameters to the outputs of stellar evolutionary models (Yi et al. 2001); the resulting values were similar.

We can also estimate the age of the star based on the rotation period. Schlaufman (2010) gave a calibrated polynomial formula relating stellar age, mass, and rotation period. Given the preceding estimates for the mass and rotation period, the Schlaufman (2010) formula gives an age of  Myr. As a consistency check we note that the stellar radius of and the rotation period of 12.5 days imply a rotation velocity of 3 km s, which is compatible with the spectroscopic estimate of (assuming ).

Radial velocities were determined via cross-correlation against synthetic templates, as described by Buchhave et al. (2010) and Quinn et al. (2012). The results are presented in Table 1, where the mean radial velocity has been subtracted. The five data points have a standard deviation of 32 m s, and internally-estimated measurement uncertainties of 20-25 m s. By fitting a sinusoid with the same period and phase as the transit signal, we obtain a velocity semiamplitude  m s. However, the true uncertainty is undoubtedly larger because of the spurious radial velocities produced by rotating starspots, which are expected to be of order 30 m s (the product of and the 1% photometric modulation). Hence we consider the data to be consistent with no radial-velocity variation. Any variation with the same period and phase as the transit signal is smaller than about 100 m s (3), corresponding to a companion mass of about 0.3  (100 ) orbiting Kepler-78.

BJD-2456300 RV (m s) RV13 (m s)
74.9484 23.66 22.62
76.9663 -36.37 19.13
81.0154 19.65 20.21
84.9219 26.03 19.13
86.9066 -32.99 25.51
Table 1Relative Radial Velocities for Kepler-78

2.3. UKIRT image

The Kepler time series is based on summing the flux within an aperture surrounding the target star Kepler-78 specific to each “season” (the quarterly 90 rotations of the field of view). The aperture dimensions change with the season; they are as large as 5 pixels (20) in the column direction and 6 pixels (24) in the row direction, with a total of 12-20 pixels used in a given season. To check whether the summed flux includes significant contributions from known neighboring stars, we examined the band image from the UKIRT survey of the Kepler field14.

Two neighboring stars were detected (see Figure 4). Relative to Kepler-78, the first neighbor is 4.5 mag fainter and 4.8 away, contributing 1.5% of the -band flux to the Kepler photometric aperture. The second neighbor is 3.1 mag fainter and 10.3 away. If this star were wholly within the aperture it would have contributed 5.5% of the total -band flux; however, since it falls near the edge of the aperture, the contribution is likely smaller and is expected to vary with the Kepler seasons. In section 4 we will show that neither of these fainter stars can be the source of the transit signal.

3. Light curve analysis

Parameter Value 68.3% Conf. Limits
KIC number 8435766
R.A. (J2000) 19h 34m 58.00s
Decl. (J2000) 44265399s
Effective temperature,  [K]15 5089
Surface gravity,  [ in cm s]16 4.60
Metallicity, [m/H]17
Projected rotational velocity,  [km s]18 2.4
Radial velocity of the star,  [km s]
Mass of the star,  []19
Radius of the star,  []20
“Spectroscopic” mean stellar density,  [g cm]21
Rotation period [days] 12.5
Reference epoch [BJD]
Orbital period [days]
Square of planet-to-star radius ratio, [ppm] ,
Planet-to-star radius ratio, ,
Scaled semimajor axis, 3.0
Orbital inclination,  [deg]
Transit duration (first to fourth contact) [hr] ,
“Photometric” mean stellar density22,  [g cm]
Seasonal dilution parameters [%]23 3.5, 0, 0.9, 5.5 1.2, 0, 1.2, 1.2
Occultation depth, [ppm] 10.5
Amplitude of illumination curve, [ppm] 4.4
Amplitude of ellipsoidal light variations, [ppm] 1.2 (3)
Planet radius,  [R]
Planet mass,  []24 8 (3)

Note. – The Kepler input catalog gives magnitudes , , , , and  K.

Table 2System Parameters of Kepler-78b

3.1. Transit times and orbital period

In the first step of the light-curve analysis we determined the orbital period , and checked for any transit-timing variations (TTV). Using the initial FT-based estimate of , we selected a 6-hour interval (12 data points) surrounding each predicted transit time. We corrected each transit interval for the starspot-induced flux modulation by masking out the transit data (the central 2 hours), fitting a linear function of time to the out-of-transit data, and then dividing the data from the entire interval by the best-fitting linear function. A total of 3378 individual transits were detected and filtered in this way; a few others were detected but not analyzed further because fewer than 2 hours of out-of-transit data are available.

\epsfxsize=3.6in \epsffilefig2.eps

Figure 2.— Deviations of individual transit times from strict periodicity.

In order to obtain precise transit times, we first need to obtain an empirical transit template. For that, the time series was folded with the trial period. As seen in Figure 1, the transit shape can be approximated by a triangular dip, due to the 30-minute averaging time of the Kepler observations. For transit timing purposes we used a triangular model, with three parameters describing the depth, duration, and time of the transit. We found the best-fitting model to the phase-folded light curve, and then fitted each individual transit with the same model, allowing only the time of transit to be a free parameter. As an estimate of the uncertainty in each data point, we used the standard deviation of the residuals of the phase-folded light curve.

The mean orbital period and a fiducial transit time were determined from a linear fit to the individual transit times, after performing 5 clipping to remove outliers. Table 2 gives the results, and Figure 2 shows the residuals. We searched the residuals for periodicities in the range 10-1000 days using a Lomb-Scargle periodogram (Scargle 1982), but found none with false alarm probability lower than 1%.

To search for any secular variation in the period, such as a period decrease due to tidal decay, we tried modeling the transit times with a quadratic function. The fit did not improve significantly. Based on this fit, the period derivative must be . Using Kepler’s third law, this can be transformed into a lower bound on the tidal decay timescale of 4 Myr. This is not a particularly interesting bound, given that the stellar age is estimated to be 750 Myr.

3.2. Transit and illumination curve analysis

\epsfxsize=5.4in \epsffilefig3.eps

Figure 3.— Upper panel.—The final light curve (dots) and best-fitting model (red curve). The inset panel is a scale illustration of the system, where the length of the dashed line represents one possible orbital distance (both the impact parameter and the orbital distance are highly uncertain). The transits look V-shaped because of the 30 minute sampling, but shorter-cadence observations should reveal a much longer flat bottom in the middle of the transit. Lower panel.—Close-up of the illumination curve and occultation. The data have been binned to 4 min for clearer visual inspection.

We then returned to the original time series and repeated the process of filtering out the starspot-induced variations, this time with the refined orbital period and a slightly different procedure. The basic idea was to filter out any variability on timescales longer than the orbital period. First, we divide the flux series on each quarter by its mean. Then, for each data point , a linear function of time was fit to all the out-of-transit data points at times satisfying . Then was replaced by , where is the best-fitting linear function. Figure 3 shows the resulting light curve, after further correcting for seasonal-specific dilution as described below.

Subsequent analysis was restricted to data from quarters 2-15, since quarters 0 and 1 had shorter durations and the data seem to have suffered more from systematic effects. For each season, we phase-folded the data with period and then reduced the data volume by averaging into 2 min samples. We then fitted a model including a transit, an occultation, and orbital phase modulations.

The transit model was based on the Mandel & Agol (2002) equations for the case of quadratic limb darkening. The parameters were the midtransit time , the zero-limb-darkening transit depth , the impact parameter , the scaled orbital separation , and the limb darkening coefficients and . The orbital period was held fixed, and a circular orbit was assumed. The limb-darkening coefficients were allowed to vary, but the difference was held fixed at 0.4, and the sum was subjected to a Gaussian prior of mean 0.7 and standard deviation 0.1. These numerical values are based on the theoretical coefficients given by Claret & Bloemen (2011).

The occultation model was a simple trapezoidal dip, centered at an orbital phase of 0.5, and with a total duration and ingress/egress durations set equal to those of the transit model. The only free parameter was the depth .

The out-of-transit modulations were modeled as sinusoids with phases and periods appropriate for ellipsoidal light variations (ELV), Doppler boosting (DB), and illumination effects (representing both reflected and reprocessed stellar radiation). Expressed in terms of orbital phase , these components are


A constant was added to the model flux, specific to each of the 4 seasons, representing light from neighboring stars. Since a degeneracy prevents all 4 constants from being free parameters, we set this “dilution flux” equal to zero for season 1, for which initial fits showed that the dilution was smallest. The other constants should be regarded as season-specific differences in dilution. The final parameter in the model is an overall flux multiplier, since only the relative flux values are significant. In the plots to follow, the normalization of the model and the data was chosen such that the flux is unity during the occultations, when only the star is seen.

For comparison with the data, the model was evaluated once per minute, and the resulting values were averaged in 29.4-min bins to match the time averaging of the Kepler data. Initial fits showed that the ELV and DB terms were consistent with zero. The non-detection of the DB term can, in principle, also be used to place upper bounds on several other effects, like inhomogeneities of the planetary albedo or a displacement of the hottest atmospheric spot on the surface of the planet with respect to the substellar point (Faigler et al. 2013). In the absence of these phenomena, the DB signal is expected to be negligible for this system ( ppm), so we set in subsequent fits. We optimized the model parameters by minimizing the standard function, and then used the best-fitting dilution parameters to correct all of the data to zero dilution.

Finally, we combined all of the dilution-corrected data to make a single light curve with 2-min sampling, and determined the allowed ranges for the model parameters using a Monte Carlo Markov Chain algorithm. The uncertainties in the flux data points were assumed to be identical and Gaussian, with magnitude set by the condition . Figure 3 shows the final light curve, with 4-min sampling, and the best-fitting model. Table 2 gives the results for the model parameters.

Some of these transit parameters might be affected by our choice for the filter. Using a filtering interval length of 2 or 3 rather than gives similar-looking light curves but with increased scatter, as expected, since the accuracy of the linear approximation for the stellar flux modulation degrades for longer time intervals. We fit these two noiser light curves with the same model and found that most of the transit parameters are not changed significantly. The secondary eclipse depth obtained were 9.8 and 10.0 ppm, slightly smaller than the value quoted on Table 2. In the case of , we obtained 4.30 and 4.65 ppm. We set the systematic error induced by the filtering to be equal to the standard deviation of the three values obtained with the three different filtering periods, and add those in quadrature to the uncertainties obtained from the MCMC routine. In both cases this procedure increased the uncertainties by only 10%.

One notable result is that the transit impact parameter is nearly unconstrained, i.e., . This is because the ingress and egress duration are poorly constrained, due to the 30 min averaging of the Kepler long-cadence data. For the same reason, is poorly determined.

We were able to place an upper limit on the ELV amplitude of 1.2 ppm (3). This can be translated into an upper bound on the mass of the transiting companion using the formula (Morris & Naftilan 1993; Barclay et al. 2012):


where is the linear limb-darkening coefficient and is the gravity-darkening coefficient. In this case we expect and (Claret & Bloemen 2011). The upper limit on the ELV amplitude thereby corresponds to a mass limit (3). (This assumes the photometric signal is entirely from the transited star, with no dilution from neighboring stars; see the following section for a discussion of possible dilution.)

The preceding analysis assumed the orbit to be circular, which is reasonable given the short orbital period and consequently rapid rate expected for tidal circularization. We can also obtain empirical constraints on the orbital eccentricity based on the timing and duration of the occultation relative to the transit. For this purpose we refitted the data adding two additional parameters, for the duration and phase of the occultation. As expected the data are compatible with a circular orbit, and give upper limits , and (3).

4. Validation as a planet

What appears to be the shallow transit of a planet may actually be the eclipse of a binary star system superposed on the nonvarying light from the brighter intended target star. The binary could be an unrelated background object, or it could be gravitationally bound to the target star and thereby be part of a triple star system. In this section we examine these possibilities, and find it more likely that the signal actually arises from the transits of a planet.

4.1. Image photocenter motion

We start with an analysis of the image photocenter location, to try to extract information about the spatial location of the varying light source (see for example Jenkins et al. 2010). For each of the four Kepler seasons, we filtered the time series of the photocenter row and column pixel coordinates in the same manner that was described in Section 3.2 for the flux time series. For each of these time series, we calculated the mean of the in-transit coordinate, the mean of the out-of-transit coordinate, and the differences between those means, which we denote and . Using the filtered flux time series for each season, we also calculated the mean of the in-transit fluxes and the mean of the out-of-transit fluxes, both normalized to the overall mean flux. We denote the difference between these two means as .

We then examined the ratios and . When either of these is multiplied by the pixel size (4), one obtains the angular offset between the varying source of light and the out-of-transit image photocenter. One can then compare these offsets to the locations of the stars revealed in the UKIRT images. Since the celestial coordinates of the signals in the Kepler aperture are difficult to obtain with high accuracy, we used the center-of-light of the three detected stars in the -band UKIRT image as our estimate for the celestial coordinates of the out-of-transit photocenter of the Kepler signal. The resulting determinations of the spatial location of the varying light source are shown in Figure 4. These results indicate that the source of the transit signal cannot be either of the two known neighboring stars. The variable source must be located within of Kepler-78, close enough to severely restrict the possibility of a blend with a background binary (Morton & Johnson 2011).

\epsfxsize=3.3in \epsffilefig4.eps

Figure 4.— Position of the three brightest stars (star symbols) in the vicinity of Kepler-78 relative to the center of light in J band (intersection of dashed lines). The positions of the source of the transits obtained from the centroid shifts (see text), represented by solid small circles with error bars, are close to the position of Kepler-78. The inset shows an expanded view of the vicinity of Kepler-78.

The inferred coordinates of the time-variable source are correlated with the Kepler seasonal dilution parameters (see Table 2), in a manner that is consistent with variable contamination by the brighter of the two neighboring stars. In season 1, when the diluting flux was found to be smallest, the centroid shift was also smallest ( arcsec). A zero centroid shift during transit implies that other sources of light have a negligible contribution to the total light within the aperture. Based on this, the brighter neighboring star can contribute no more than 1-2% of the total flux within the aperture during season 1. This supports our choice of a dilution parameter equal to zero for season 1.

This analysis of the photocenter motion is sufficient for our purposes. However, as pointed out by Bryson et al. (2013), for a more detailed analysis it would be advisable to fit the pixel data using the Kepler Pixel Response Function, rather than using moment-based centroids as we have done.

4.2. Ellipsoidal light variations

The photocenter analysis cannot rule out the possibility that the observed phenomena are due to a background eclipsing binary gravitationally bound to the target star in a triple system or coincidentally within of the target star. In such cases, though, the ELVs would generally be larger than the observed upper bound, as we show presently. This technique was recently used by Quintana et al. (2013) to validate a hot Jupiter.

In Section 3.2 we found that the upper limit on leads to an upper limit on the companion mass of . This analysis was based on the assumption that the dilution was small. If the signal arises from a faint unresolved eclipsing binary, the ELV signal could have been diluted by a large factor, weakening the constraint on the companion mass. However, an upper bound on the dilution factor can be obtained from further analysis of the folded light curve. This can be understood qualitatively as follows. If the dilution is severe, then the true transit/eclipse is much deeper than 200 ppm and the ratio of the radii of the secondary and primary star must be larger than the ratio inferred assuming no dilution. A large radius ratio implies relatively long ingress and egress durations, which at some point become incompatible with the observed light curve shape. The sensitivity of this test is hindered by the 29.4 min cadence of the Kepler data, but in this case it still provides a useful constraint.

For a quantitative analysis we reanalyzed the phase-folded light curve in a manner similar to that presented in Section 3.2. We suppose that the observed flux is , where is the undiluted transit/occultation signal and is the diluting flux. Since our model is always normalized to have unit flux during the occultation, we use units in which during the occultation, and we also defined a normalized version of ,


With this definition , where is the observed depth of the transit and is the actual depth before dilution. We stepped through a range of values for , finding the best-fitting model in each case and recording the goodness-of-fit . We found with 3 confidence.

In the presence of dilution, Eqn. (2) can be used to solve for the mass ratio,


using notation that does not presume the secondary is a planet ( and are the secondary and primary masses, and is the primary radius). An upper limit on the mass ratio follows from the constraints ,  ppm, and . The latter constraint follows from the requirement that the maximum eclipse duration of exceed the observed eclipse duration ; the factor of 1.25 arises from the fact that at the maximum possible dilution the secondary star is approximately 1/4 the size of the primary star. The result is


Having established that regardless of dilution, we may obtain a second condition on the mass ratio by using Kepler’s third law, , to eliminate from Eqn. (4). This gives


which leads to another inequality,


The scaling with makes clear that the most massive secondaries are allowed for small primary stars, i.e., main-sequence stars and not giants. On the lower main sequence,


  1. affiliation: Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;,,
  2. affiliation: Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;,,
  3. affiliation: Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;,,
  4. affiliation: M.I.T. Kavli Institute for Astrophysics and Space Research, 70 Vassar St., Cambridge, MA, 02139;
  5. affiliation: Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
  6. affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA, 02138, USA
  7. affiliation: Niels Bohr Institute, University of Copenhagen, Juliane Maries vej 30, 2100 Copenhagen, Denmark
  8. affiliationmark:
  9. journal: Astrophysical Journal, in press
  10. slugcomment: Accepted for publication, 2013 July 11
  11. affiliationtext: Centre for Star & Planet Formation, Natural History Museum of Denmark, University of Copenhagen, Øster Voldgade 5-7, 1350 Copenhagen, Denmark
  13. Estimated 1- uncertainty in the relative radial velocity.
  15. Obtained from an SPC analysis of the spectra.
  16. Obtained from an SPC analysis of the spectra.
  17. Obtained from an SPC analysis of the spectra.
  18. Obtained from an SPC analysis of the spectra.
  19. Based on the relationships from Torres & Andersen (2010).
  20. Based on the relationships from Torres & Andersen (2010).
  21. Based on the relationships from Torres & Andersen (2010).
  22. Defined as
  23. In order, these refer to season 0, 1, 2 and 3.
  24. Based on absence of ellipsoidal light variations, assuming zero dilution.
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